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We are given that
(dy)/(dx)=x^(2)-2y.
Fasind an expression for
(d^(2)y)/(dx^(2)) in terms of
x and
y.

(d^(2)y)/(dx^(2))=

We are given that $\frac{d y}{d x}=x^{2}-2 y$ . $\newline$ Fasind an expression for $\frac{d^{2} y}{d x^{2}}$ in terms of $x$ and $y$ . $\newline$ $\frac{d^{2} y}{d x^{2}}=$

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Find the vertex of the transformed function

Full solution

Q. We are given that

\frac{d y}{d x}=x^{2}-2 y

.

\newline

Fasind an expression for

\frac{d^{2} y}{d x^{2}}

in terms of

x

and

y

.

\newline

\frac{d^{2} y}{d x^{2}}=

Given Derivative Equation: We are given the first derivative of $y$ with respect to $x$ as $\frac{dy}{dx}=x^{2}-2y$ . To find the second derivative $\frac{d^{2}y}{dx^{2}}$ , we need to differentiate $\frac{dy}{dx}$ with respect to $x$ .
Apply Chain Rule: Differentiate $\frac{dy}{dx}$ with respect to $x$ using the chain rule for the term $-2y$ , since $y$ is a function of $x$ . This means we need to multiply the derivative of $-2y$ with respect to $y$ , which is $-2$ , by the derivative of $y$ with respect to $x$ , which is $\frac{dy}{dx}$ .
Calculate Second Derivative: The derivative of $x^{2}$ with respect to $x$ is $2x$ . The derivative of $-2y$ with respect to $y$ is $-2$ , and we multiply this by $(dy)/(dx)$ to apply the chain rule. So, the second derivative $(d^{2}y)/(dx^{2})$ is $2x - 2(dy)/(dx)$ .
Substitute and Simplify: Substitute the expression for $\frac{dy}{dx}$ from the given equation into the expression for $\frac{d^{2}y}{dx^{2}}$ . This gives us $\frac{d^{2}y}{dx^{2}} = 2x - 2(x^{2} - 2y)$ .
Final Second Derivative Expression: Simplify the expression for $(d^{2}y)/(dx^{2})$ by distributing the $-2$ and combining like terms. This gives us $(d^{2}y)/(dx^{2}) = 2x - 2x^{2} + 4y$ .
Final Second Derivative Expression: Simplify the expression for $(d^{2}y)/(dx^{2})$ by distributing the $-2$ and combining like terms. This gives us $(d^{2}y)/(dx^{2}) = 2x - 2x^{2} + 4y$ .The final expression for the second derivative of $y$ with respect to $x$ in terms of $x$ and $y$ is $(d^{2}y)/(dx^{2}) = -2x^{2} + 2x + 4y$ .

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